Cyclic extensions of $K(\sqrt {-1})/K$

Abstract:

In this paper the height ${\text {ht}}(L/K)$ of a cyclic $2$-extension of a field $K$ of characteristic $\ne 2$ is studied. Here ${\text {ht}}(L/K) \geq n$ means that there is a cyclic extension $E$ of $K,E \supset L$, with $[E:L] = {2^n}$. Necessary and sufficient conditions are given for ${\text {ht}}(L/K) \geq n$ provided $K(\sqrt { - 1})$ contains a primitive ${2^n}$th root of unity. Primary emphasis is placed on the case $L = K(\sqrt { - 1})$. Suppose ${\text {ht}}(K(\sqrt { - 1})/K) \geq 1$. It is shown that ${\text {ht}}(K(\sqrt { - 1})/K) \geq 2$ and if $K$ is a number field then ${\text {ht}}(K(\sqrt { - 1})/K) \geq n$ for all $n$. For each $n \geq 2$ an example is given of a field $K$ such that ${\text {ht}}(K(\sqrt { - 1})/K) \geq n$ but ${\text {ht}}(K(\sqrt { - 1})/K) \ngeq n + 1$.